Shallow Water Wave Theory Research Articles

An experimental investigation of roll waves in high pressure two-phase inclined pipe flows

This paper examines the effects of small upward inclinations on the formation of roll waves and the properties of fully developed roll waves at high pressure conditions. A total of 984 experiments were conducted at six positive pipe inclinations θ = 0.00°, 0.10°, 0.25°, 1.00°, 2.50° and 5.00° using a 25 m long 10 cm i.d. pipe. Sulfur hexafluoride (SF 6) was used at 8 bara giving a gas density of 50 kg/m 3. Two independent mechanisms for the formation of roll waves were identified; (1) interaction between 2D shallow water waves and (2) a visible long wavelength instability of the stratified layer. Viscous long wavelength linear stability analysis predicted the critical liquid flow rate and liquid height for the initiation of roll waves when roll waves were formed due to the second mechanism. A simple equation from shallow water wave theory agreed with measurements for critical liquid flow rate when roll waves were formed due to the first mechanism. Shallow water wave speed agreed with critical wave speeds at transition and nonlinear wave speeds for fully developed roll waves in certain cases. The increase in interfacial friction due to the presence of large waves was compared with models from the literature.

Effect of Tsunamis generated in the Manila Trench on the Gulf of Thailand

Tsunamis generated in the Manila Trench can be a threat to Thailand. Besides runup of tsunamis along the eastern coast, infrastructures in the Gulf of Thailand, for example, gas pipelines and platforms can be affected by tsunamis. In this study, the simulation of tsunamis in the Gulf of Thailand is conducted. Six cases of fault ruptures in the Manila trench are considered for earthquakes with magnitudes of 8.0, 8.5, and 9.0. The linear shallow water wave theory in spherical coordinate system is used for tsunami simulation in the large area covering Southeast Asia while the nonlinear shallow water wave theory in Cartesian coordinate system is used for tsunami simulation in the Gulf of Thailand. It is found that tsunamis reach the southern part of Thailand in 13 h after an earthquake and reach Bangkok in 19 h. The tsunami amplitude is largest in the direction towards the Philippines and Vietnam. The southern part of China is also severely affected. The Gulf of Thailand is affected by the diffraction of tsunamis around the southern part of Vietnam and Cambodia. The tsunami amplitude at the southernmost coastline is about 0.65 m for the M w 9.0 earthquake. The current velocity in the Gulf of Thailand due to the M w 9.0 earthquake is generally less than 0.2 m/s.

A Toy Model of the Instability in the Equatorially Trapped Convectively Coupled Waves on the Equatorial Beta Plane

Abstract The equatorial atmospheric variability shows a spectrum of significant peaks in the wavenumber–frequency domain. These peaks have been identified with the equatorially trapped wave modes of rotating shallow water wave theory. This paper addresses the observation that the various wave types (e.g., Kelvin, Rossby, etc.) and wavenumbers show differing signal strength relative to a red background. It is hypothesized that this may be due to variations in the linear stability of the atmosphere in response to the various wave types depending on both the specific wave type and the wavenumber. A simple model of the convectively coupled waves on the equatorial beta plane is constructed to identify processes that contribute to this dependence. The linear instability spectrum of the resulting coupled system is evaluated by eigenvalue analysis. This analysis shows unstable waves with phase speeds, growth rates, and structures (vertical and horizontal) that are broadly consistent with the results from observations. The linear system, with an idealized single intertropical convergence zone (ITCZ) as a mean state, shows peak unstable Kelvin waves around zonal wavenumber 7 with peak growth rates of ∼0.08 day−1 (e-folding time of ∼13 days). The system also shows unstable mixed Rossby–gravity (MRG) and inertio-gravity waves with significant growth in the zonal wavenumber range from −15 (negative indicates westward phase speed) to +10 (positive indicates eastward phase speed). The peak MRG n = 0 eastward inertio-gravity wave (EIG) growth rate is around one-third that of the Kelvin wave and occurs at zonal wavenumber 3. The Rossby waves in this system are stable, and the Madden–Julian oscillation is not observed. Within this model, it is shown that in addition to the effect of the ITCZ configuration, the differing instabilities of the different wave modes are also related to their different efficiency in converting input energy into divergent flow. This energy conversion efficiency difference is suggested as an additional factor that helps to shape the observed wave spectrum.

The reduction effects of mangrove forest on a tsunami based on field surveys at Pakarang Cape, Thailand and numerical analysis

Using an integrated approach including satellite imagery analysis, field measurements, and numerical modeling, we investigated the damage to mangroves caused by the 2004 Indian Ocean tsunami at Pakarang Cape in Pang Nga Province, Thailand. Comparing pre- and post-tsunami satellite imagery of the study area, we found that approximately 70% of the mangrove forest was destroyed by the tsunami. Based on field observations, we found that the survival rate of mangroves increased with increasing stem diameter. Specifically, we found that 72% of Rhizophora trees with a 25–30 cm stem diameter survived the tsunami impact, whereas only 19% with a 15–20 cm stem diameter survived. We simulated the 2004 Indian Ocean tsunami using the nonlinear shallow-water wave theory to reproduce the tsunami inundation flow and investigated the bending moment acting on the mangrove trees. Results of the numerical model showed that the tsunami inundated areas along the mangrove creeks, and its current velocity reached 5.0 m s −1. Based on the field measurements and numerical results, we proposed a fragility function for mangroves, which is the relationship between the probability of damage and the bending stress caused by the maximum bending moment. We refined the numerical model to include the damage probability of mangrove forests using the obtained fragility function to investigate the tsunami reduction effect of mangrove forest. Under simple numerical conditions related to the mangrove forest, ground level, and incident wave, the model showed that a mangrove forest of Rhizophora sp. with a density of 0.2 trees m −2 and a stem diameter of 15 cm in a 400 m wide area can reduce the tsunami inundation depth by 30% when the incident wave is assumed to have a 3.0 m inundation depth and a wave period of 30 min at the shoreline. However, 50% of the mangrove forest is destroyed by a 4.5 m tsunami inundation depth, and most of the mangrove forest is destroyed by a tsunami inundation depth greater than 6 m. The reduction effect of tsunami inundation depth decreased when the tsunami inundation depth exceeded 3 m, and was mostly lost when the tsunami inundation depth exceeded 6 m.

Analysis of Origin of Multi-Ring Basins by Theory of Deep Water Waves

The tsunami model of the origin of multi-ring basins is analysed with the theory of deep water waves generated by an initial surface deformation, which is set as a parabolic crater. We obtain an approximate formula for calculating the ring radius. The formula applied to some multi-ring basins on the Moon, Mercury and Mars gives almost equidistant spacing of the rings within the main ring (the IV ring); this agrees with the previous conclusion that the IV ring marks the end of the fluidized region. Besides this, the theory of deep water waves does not require similar crustal structure at each basin-impact site on all three planets which is required in the theory of shallow water waves.

Free surface water wave 1-D LBGK predictions

The performance of a single phase free surface lattice Boltzmann (LB) solver is investigated. The lattice-BGK (LBGK) model explores the free surface behavior with a focus on non-overturning water waves. The discretized equations are based on shallow water wave theory in viscous flow. Therefore, only flows with horizontal velocity dominating free surface waves is applicable. The solutions on uniform grids implement an elastic-collision scheme assuming slip boundaries at the tank walls. Simulations are carried out in horizontal base-excited square tanks. The forcing amplitude and the tank aspect ratios of are kept constant while the forcing frequency is varied off- and near resonance. At this water depth, the free surface was observed to slosh in a first sway mode initially; however, the following cycles would typically consist of traveling waves or bores. The flow typically remained 1-D in resonance cases except at the bore front and near the walls. The LB experiments are limited to flow simulations of weak bores which occurred near resonance with forcing amplitude in the order of . The LB solutions were in good agreement with a Riemann solver and agreed fairly well with experimental data.

The near-field tsunami amplitudes caused by submarine landslides and slumps spreading in two orthogonal directions

The aim of this study is to present the solutions for the near-field tsunami amplitudes caused by submarine landslumps and slides spreading in two orthogonal directions. A linearized shallow water wave theory is derived. The transform techniques (Fourier and Laplace transform) are used for the solution of Laplace equation. The results show that if the ratio of the velocties is v 1/ v 2=0.1, the numerical results are almost the same as the values obtained for one dimensional movement of the slumps and slides. But, when the ratio of the velocties is v 1/ v 2=1, obtained normalized peak amplitudes, η max/ ζ 0 are smaller than the numerical values for one dimensional solution. It is concluded that normalized peak amplitudes for the models are small because of the interaction of the velocities. Numerical examples are presented for various parameters.

The Diffraction of Multidirectional Random Waves by Rectangular Submarine Pits

The diffraction of multidirectional random surface waves with one or more rectangular submarine pits is investigated theoretically. The incident wave conditions are specified using a discrete form of the Mitsuyasu directional spectrum. The numerical model involves the superposition of regular-wave diffraction solutions based on linearized shallow water wave theory obtained by a two-dimensional boundary integral approach for water of uniform depth. Numerical results are presented for multi-directional random waves that illustrate the effect of the various wave and pit parameters on the diffraction characteristics of typical single and multiple pit systems.

Ocean depth effects on tsunami amplitudes used in source models in linearized shallow-water wave theory

The aim of this study is to determine the effects of ocean depth on tsunami amplitudes used in source models for use in linear shallow-water wave theory. To determine the effects, 19 ocean depths ranging from 200–2000 m are studied. The results show that it is possible to establish a series-type exploration function to define the depth-amplitude effects. This function with the unknown coefficients related to ruptures of the ocean floor involves the normalized maximum tsunami amplitude. Results obtained from this series-function give good approximations for four or more terms in the series. Two basic source models are considered. Numerical examples are presented.

Tsunami run-up and draw-down on a plane beach

Tsunami run-up and draw-down motions on a uniformly sloping beach are evaluated based on fully nonlinear shallow-water wave theory. The nonlinear equations of mass conservation and linear momentum are first transformed to a single linear hyperbolic equation. To solve the problem with arbitrary initial conditions, we apply the Fourier–Bessel transform, and inversion of the transform leads to the Green function representation. The solutions in the physical time and space domains are then obtained by numerical integration. With this semi-analytic solution technique, several examples of tsunami run-up and draw-down motions are presented. In particular, detailed shoreline motion, velocity field, and inundation depth on the shore are closely examined. It was found that the maximum flow velocity occurs at the moving shoreline and the maximum momentum flux occurs in the vicinity of the extreme draw-down location. The direction of both the maximum flow velocity and the maximum momentum flux depend on the initial waveform: it is in the inshore direction when the initial waveform is predominantly depression and in the offshore direction when the initial waves have a dominant elevation characteristic.

Diffraction of multidirectional random waves by multiple rectangular submarine pits

A numerical model is presented to predict the interaction of multidirectional random surface waves with one or more rectangular submarine pits. The water depth is assumed uniform and the method involves the superposition of diffraction solutions based on linearized shallow water wave theory obtained by a two-dimensional boundary integral approach. The incident wave conditions are specified using a discrete form of the Mitsuyasu directional spectrum. The present numerical model has been validated through comparisons with previous theoretical results for regular waves. Good agreement was obtained in all cases. Based on these comparisons it is concluded that the present numerical model is an accurate and efficient tool to predict the wave field around multiple submarine pits and navigation channels in many practical situations.

Helium Detonations on Neutron Stars

We present the results of a numerical study of helium detonations on the surfaces of neutron stars. We describe two-dimensional simulations of the evolution of a detonation as it breaks through the accreted envelope of the neutron star and propagates laterally through the accreted material. The detonation front propagates laterally at nearly the Chapman-Jouguet velocity, v = 1.3 × 109 cm s-1. A series of surface waves propagate across the pool of hot ash behind the detonation front with the same speed, matching the speed expected from shallow water wave theory. The entire envelope oscillates in the gravitational potential well of the neutron star with a period of ~50 μs. The photosphere reaches an estimated height of 10 km above the surface of the neutron star. Our study confirms that such a detonation can insure the spread of burning over the entire neutron star surface on a timescale consistent with burst rise times. We analyze the sensitivity of the results to the spatial resolution and the assumed initial conditions. We conclude by presenting a comparison of this model to type I X-ray bursts.

Bedforms produced by impact-generated tsunami, ∼2.6 Ga Hamersley basin, Western Australia

Abstract Late Archean strata in the Hamersley basin of Western Australia contain a layer of anorbital wave ripples composed largely of impact-generated melt spherules. Sedimentologic evidence indicates that these ripples formed in a deep marine shelf setting well below the effects of meteorological processes. Combined with the impact-related origin of the spherules, this suggests that the ripples were formed by an impact-generated tsunami. To test if this geologic interpretation is reasonable, we have modeled anorbital ripple formation by impact-generated tsunami waves in the Hamersley basin. We assume a deep ocean target area, and that tsunami waves traveled from the target area unhindered to the Hamersley basin. Using melt spherule radius as an indicator, we infer a bolide diameter of 4.9 km. Assuming a bolide density of 2.6 and a velocity of 20 km/s, this impact would produce a crater 60.4 km in diameter. We modeled tsunami waves generated by the collapse of this crater, and propagated them into the Hamersley basin using shallow water wave theory. The wave models indicate tsunami waves created at impacts between 2000 and 8000 km from the Hamersley basin could have generated the observed anorbital ripples.

Hydraulic Sorting of Heavy-Mineral Grains by Swash on a Medium-Sand Beach

ABSTRACT Hydraulic sorting of detrital mineral grains in the swash zone was investigated using data on sedimentology and flow dynamics obtained from Fishermans Beach, on the east coast of Australia. The beach is characteristically reflective, displaying a steep beach face composed of medium sand and virtually no surf zone. Samples were taken from two beds enriched in heavy minerals, and from the adjacent beds above and below. The mineralogy of the samples was determined, and measurements of grain size and settling velocity were obtained. The heavy-mineral grains had both a smaller intermediate diameter and a smaller settling velocity than the light-mineral grains; the difference between the two mineral types was typically in the range 0.25-0.50 and 2.0-2.5 cm s-1, respectively. On the basis of these sedimentological measurements alone, any one of three mineral sorting mechanisms could provide a feasible explanation for the development of the heavy-mineral-enriched beds: suspension sorting, entrainment sorting, or shear sorting. A numerical model for swash, based on the nonlinear shallow-water wave theory, is used to provide a quantitative description of flow dynamics in the swash zone at the time the heavy-mineral-enriched beds were formed. For the probable wave height responsible for producing the enriched beds, modeled maximum flow velocity and bed shear stress at the mid-swash position reached 3.5 m s-1 and 79.51 dynes cm-2, respectively. Given the shallow water depths and the lack of sorting in the horizontal direction of flow, suspension sorting does not seem to be of overriding importance at Fishermans Beach. The large bed shear stresses seem to also preclude entrainment sorting, because for most of the time all mineral types are predicted to be in motion. This contrasts with previous studies on finer-grained beaches, where entrainment sorting was favored, and suggests that beach type may play a role in determining the sorting mechanism operating at a particular location. Shear sorting was found to be feasible within the modeled flow constraints at Fishermans Beach, but it does not provide an entirely satisfactory explanation for the observed mineral sorting patterns. We contend that the inability to unequivocally identify the true sorting mechanism is more a reflection of our poor understanding of the operation of these sorting mechanisms, particularly within the swash zone, rather than an indication that any of these mechanisms are physically inoperative in the swash zone.

Modeling of Tuned Liquid Damper With Submerged Nets

An analytical model for describing the effectiveness of tuned liquid damper (TLD) with submerged nets for suppressing horizontal vibration of structures is proposed. Dissipation energy due to the liquid motion under sinusoidal excitation is calculated based on nonlinear shallow water wave theory. In particular, the effects of hydraulic resistance produced by nets installed in a tank on the performance of TLDs are examined. The results of dissipation energy theoretically obtained are confirmed by experiments. To show the effectiveness of the proposed analytical model for TLD-structure interaction problems, the case in which TLD with nets is installed on top of a pylon of a cable-stayed bridge is described and the calculated results are compared with the actual experimental data. In the calculation, it is shown that the optimal damping factor, as is the case for tuned mass dampers (TMD), can be produced by the nets, and the effectiveness of TLD with nets are demonstrated.

Propagation of Obliquely Incident Tsunamis on a Slope Part II Characteristics of on-Ridge Tsunamis

Fundamental characteristics of tsunamis propagating over an oceanic ridge are investigated analytically based on the linear shallow water wave theory. The oceanic ridge model used has a triangular shape which is formed by the combination of a constant slope and a bottom of uniform depth. Wave pattern of on-ridge tsunamis varies depending on the obliquity of incident waves. Especially, standing waves are formed when the incident wave angle to the ridge exceeds the critical angle. Then no transmitted waves exist because of the evanescent waves. The present study is applied to the 1996 Irian Jaya tsunami which propagated over the South-Honshu ridge. The characteristics of on-ridge tsunamis observed at Chichijima and Hachijojima, which are close to the South-Honshu ridge, can be partly explained by using the solution derived in the present study.

Propagation of Obliquely Incident Tsunamis on a Slope Part I: Amplification of Tsunamis on a Continental Slope

A new analytical solution of tsunami which is obliquely incident on a continental slope is derived based on the linear shallow water wave theory. The model of a continental slope consists of a finite constant slope and a bottom of uniform depth and has a vertical wall at the shoreline. Because of the existence of the vertical wall, another function can be taken as one of the fundamental solutions on a slope in addition to the confluent hypergeometric function M. Based on the solution obtained, the characteristics of tsunami amplification on a continental slope are discussed in terms of the conditions of the incidence. The magnitude of the amplification can be predicted with several parameters of wave incidence, such as, the incident angle and the relative incident-wave length to the topography. Especially, when a tsunami is incident almost parallel to the slope, we found that there exists distinctive amplifications for particular parameters characterizing the incident wave. The effect of the existence of the vertical wall is also discussed. The characteristics of amplification also depend much on the position of the vertical wall.

Zero wave resistance for ships moving in shallow channels at supercritical speeds

This paper deals with the wave pattern and wave resistance of a slender ship moving steadily at supercritical speed in a shallow water channel. Using, successively, linear and nonlinear shallow-water wave theory it is demonstrated that, if the hull form is adapted to speed and channel geometry according to certain rules, the ship waves can be made to form a localized pattern around the ship that moves at the same speed as the ship and at the same time the associated wave resistance can be made to vanish. In the nonlinear case, the zero-wave-resistance ship hull is derived from a KP equation solution of the oblique interaction of two identical solitons. This astonishing phenomenon may be called shallow-channel superconductivity.

Scattering of long waves around a circular island mounted on a conical shoal

Scattering of simple harmonic long waves by a cylindrical island mounted on a conical shoal in an otherwise open sea of constant depth is solved analytically based on the shallow-water wave (long-wave) theory. The new analytical solution not only confirms some conclusions and conjectures previously drawn from purely numerical studies, such as those showing how the slope of the shoal affects the amplification of the ocean waves around the coastline of islands, but also provides another useful check for numerical model developers.

Multiple-Pit Breakwaters

Linearized shallow-water wave theory is used to investigate the interaction of surface waves with multiple rectangular submarine pits in water of otherwise uniform depth. The solution is obtained by a boundary element technique using a two-dimensional Green's function. It is shown that appropriate selection of pit dimensions and placement may lead to a significant reduction in wave heights behind these structures. Numerical results have been presented that illustrate the influence of the various pit characteristics on the diffracted wave field. Two pits can provide a shadow region in which wave heights are reduced to 10–20% of the incident wave height. The shadow region, with wave heights reduced to 30% of the incident wave height, is approximately the width of the pits and five wavelengths long. Guidance is given on how to select pit geometries and placement for optimal breakwater performance. An example is also presented that shows how to select a pit to reduce wave heights in a navigation channel.